Finite Group Databasebeta

← #20279192

Subgroups — #20279192

Order 6784.

544 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133193C211yesmaximal, characteristic
2C2×C4⋊C4×D53C211yesmaximal, characteristic
3#16810557C214yesmaximal, complemented
4#16810530C211yesmaximal, characteristic, complemented
5#16810497C214yesmaximal
6#16810493C211yesmaximal, characteristic, complemented
7#16810492C211yesmaximal, characteristic, complemented
8C4×#11719921C211yesmaximal, characteristic
9C53×#391C211yesmaximal, characteristic, complemented
10C22×C4×D53C2211yescharacteristic
11C22×C4×D53C2212yes—
12#13133193—24no—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×C4⋊Dic53C2211yescharacteristic
15C2×C4⋊Dic53C2214yes—
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2214yes—
19C4⋊C4×D53—24no—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×C4×Dic53C2211yescharacteristic
23#13133124C2214yes—
24#13133124C2212yes—
25C22×Q8×C53C2211yescharacteristic
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×C42×C53C2211yescharacteristic
28#95×C53C2214yes—
29C23×D53C2311yescharacteristic
30C2×Q8×C53—22no—
31C2×Q8×C53—22no—
32C2×Q8×C53—22no—
33C22×C4×C53C2312yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2311yescharacteristic
40C22×Dic53C2312yes—
41C2×C4×D53—22no—
42C2×C4×D53—22no—
43C2×C4×D53—24no—
44C2×C4×D53—44no—
45C2×C4×D53D414yes—
46C4⋊C4×C53—22no—
47C42×C53—22no—
48#11719921—24no—
49#11719921Q818yescomplemented
50C4⋊Dic53—22no—
51C4⋊Dic53—22no—
52C4⋊Dic53—44no—
53#11719919—24no—
54#11719919—24no—
55#11719919—44no—
56C4×Dic53—22no—
57C23×C53C2411yescharacteristic
58C22×D53—21no—
59C22×D53—21no—
60C22×D53—22no—
61C22×D53C2×D412yes—
62C22×D53C2×Q814yes—
63Q8×C53—44no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—24no—
71C2×C4×C53C2×D412yes—
72C2×C4×C53C2×Q814yes—
73C2×C4×C53#4112yes—
74C2×C4×C53#4112yes—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—24no—
82C2×Dic53—24no—
83C2×Dic53C2×D412yes—
84C2×Dic53C2×Q814yes—
85C4×D53—28no—
86C4×D53—48no—
87C22×C53C22×D411yescharacteristic
88C22×C53C22×Q812yes—
89C22×C53C2×#4111yescharacteristic
90C22×C53C2×#4111yescharacteristic
91C22×C53H(5,𝔽2)12yes—
92C2×D53—22no—
93C2×D53—22no—
94C2×D53—24no—
95C2×D53—24no—
96C2×D53C4⋊Q814yes—
97C4×C53—24no—
98C4×C53—42no—
99C4×C53—42no—
100C4×C53#12314yes—
101Dic53—22no—
102Dic53—24no—
103Dic53—42no—
104Dic53—42no—
105#2502—531nomaximal, Sylow
106C2×C53C2×#12311yescharacteristic
107C2×C53C2×C4⋊Q811yescharacteristic
108C2×C53#53511yescharacteristic
109C2×C53#55412yes—
110C2×C53#55412yes—
111D53—24no—
112C2×C22⋊Q8—531no—
113C22×C4⋊C4—531no—
114#398—534no—
115#391—531no—
116#391D5311yescharacteristic, complemented
117#385—534no—
118#384—531no—
119#384—531no—
120C4×C22⋊C4—531no—
121C53#250211yescharacteristic, complemented, Sylow
122C22×Q8C2×D5311yescharacteristic
123C23×C4—531no—
124C23×C4—532no—
125C22⋊Q8—1064no—
126C2×C4⋊C4—532no—
127C2×C4⋊C4—531no—
128C2×C4⋊C4—532no—
129C2×C4⋊C4—531no—
130C2×C4⋊C4—534no—
131C2×C4⋊C4—534no—
132C2×C4⋊C4—1064no—
133C2×C4⋊C4C2×D5311yescharacteristic
134C2×C22⋊C4—532no—
135C2×C22⋊C4—532no—
136C2×C42—531no—
137C2×C42C2×D5311yescharacteristic
138#95—532no—
139#95—534no—
140#95C2×D5314yes—
141C24—531no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—22no—
145C22×C4—532no—
146C22×C4—531no—
147C22×C4—534no—
148C22×C4—532no—
149C22×C4—532no—
150C22×C4—1064no—
151C22×C4—1062no—
152C22×C4—1062no—
153C22×C4—2124no—
154C22×C4C22×D5312yes—
155C22×C4C22×D5312yes—
156C22×C4C22×D5312yes—
157C22×C4C22×D5311yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—1064no—
160C4⋊C4—1064no—
161C4⋊C4—1062no—
162C4⋊C4—1062no—
163C4⋊C4—2124no—
164C4⋊C4—2124no—
165C22⋊C4—538no—
166C22⋊C4—1064no—
167C42—22no—
168C42—1062no—
169C23—532no—
170C23—534no—
171C23—1061no—
172C23—1061no—
173C23—1062no—
174C23C23×D5311yescharacteristic
175Q8—44no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—532no—
184C2×C4—534no—
185C2×C4—1061no—
186C2×C4—1062no—
187C2×C4—1061no—
188C2×C4—1062no—
189C2×C4—1068no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1064no—
193C2×C4—1064no—
194C2×C4—2128no—
195C2×C4#1171994512yes—
196C2×C4D4×D5312yes—
197C2×C4#1171994712yes—
198C2×C4Q8×D5314yes—
199C22—534no—
200C22—1064no—
201C22—1064no—
202C22—1062no—
203C22—1062no—
204C22C2×D4×D5311yescharacteristic
205C22C2×Q8×D5312yes—
206C22C2×#1171994711yescharacteristic
207C22C2×#1171994511yescharacteristic
208C22#1313326312yes—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—1064no—
213C4—1062no—
214C4—2122no—
215C4—2122no—
216C4#1313319914yes—
217C2—1064no—
218C2C2×#1313319911yescharacteristic
219C2C4⋊Q8×D5311yescharacteristic
220C2#1681119111yescharacteristic
221C2#1681133112yes—
222C2#1681133312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#391C2
hypercenter#391D53
hypocenterC53#2502
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #16810557C2semidirect
4. #16810530C2semidirect
6. #16810493C2semidirect
7. #16810492C2semidirect
9. C53×#391C2semidirect
49. #11719921Q8semidirect
116. #391D53semidirect
121. C53#2502semidirect